| L(s) = 1 | + (1.91 + 0.0910i)2-s + (1.65 + 0.509i)3-s + (1.65 + 0.158i)4-s + (−1.87 + 1.78i)5-s + (3.11 + 1.12i)6-s + (0.424 − 2.20i)7-s + (−0.637 − 0.0916i)8-s + (2.48 + 1.68i)9-s + (−3.74 + 3.24i)10-s + (−4.26 − 2.19i)11-s + (2.66 + 1.10i)12-s + (3.61 − 0.696i)13-s + (1.01 − 4.17i)14-s + (−4.01 + 2.00i)15-s + (−4.47 − 0.862i)16-s + (0.858 − 1.87i)17-s + ⋯ |
| L(s) = 1 | + (1.35 + 0.0643i)2-s + (0.955 + 0.294i)3-s + (0.827 + 0.0790i)4-s + (−0.838 + 0.799i)5-s + (1.27 + 0.459i)6-s + (0.160 − 0.832i)7-s + (−0.225 − 0.0324i)8-s + (0.826 + 0.562i)9-s + (−1.18 + 1.02i)10-s + (−1.28 − 0.662i)11-s + (0.768 + 0.319i)12-s + (1.00 − 0.193i)13-s + (0.270 − 1.11i)14-s + (−1.03 + 0.517i)15-s + (−1.11 − 0.215i)16-s + (0.208 − 0.455i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 - 0.381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.924 - 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.47201 + 0.490237i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.47201 + 0.490237i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.65 - 0.509i)T \) |
| 23 | \( 1 + (0.573 - 4.76i)T \) |
| good | 2 | \( 1 + (-1.91 - 0.0910i)T + (1.99 + 0.190i)T^{2} \) |
| 5 | \( 1 + (1.87 - 1.78i)T + (0.237 - 4.99i)T^{2} \) |
| 7 | \( 1 + (-0.424 + 2.20i)T + (-6.49 - 2.60i)T^{2} \) |
| 11 | \( 1 + (4.26 + 2.19i)T + (6.38 + 8.96i)T^{2} \) |
| 13 | \( 1 + (-3.61 + 0.696i)T + (12.0 - 4.83i)T^{2} \) |
| 17 | \( 1 + (-0.858 + 1.87i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (0.409 - 0.187i)T + (12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (0.657 + 6.88i)T + (-28.4 + 5.48i)T^{2} \) |
| 31 | \( 1 + (2.87 + 2.25i)T + (7.30 + 30.1i)T^{2} \) |
| 37 | \( 1 + (-1.58 - 5.39i)T + (-31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (-6.57 - 6.89i)T + (-1.95 + 40.9i)T^{2} \) |
| 43 | \( 1 + (-3.91 - 4.98i)T + (-10.1 + 41.7i)T^{2} \) |
| 47 | \( 1 + (8.90 - 5.13i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.890 + 1.02i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-0.375 - 1.94i)T + (-54.7 + 21.9i)T^{2} \) |
| 61 | \( 1 + (5.07 + 12.6i)T + (-44.1 + 42.0i)T^{2} \) |
| 67 | \( 1 + (-0.445 - 0.864i)T + (-38.8 + 54.5i)T^{2} \) |
| 71 | \( 1 + (1.01 - 1.57i)T + (-29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (5.00 + 10.9i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-6.31 - 2.18i)T + (62.0 + 48.8i)T^{2} \) |
| 83 | \( 1 + (7.37 + 7.03i)T + (3.94 + 82.9i)T^{2} \) |
| 89 | \( 1 + (-2.09 - 14.5i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-3.16 + 0.768i)T + (86.2 - 44.4i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.99373059468842633690168612353, −11.41284242041275420891161619438, −10.87321622830654033362068759463, −9.612149755564258625341386651914, −8.060299666138439331995635162097, −7.49761830916720679324481951334, −6.06376928490764533366149660619, −4.62874605036612706994850037855, −3.62054303068549525907401127118, −2.93725705810328785284474942726,
2.31912401035301126267722924100, 3.65976752464577311816621559384, 4.63947972030040267410664476081, 5.74419366084488535385577492090, 7.24628901772808264191560927397, 8.462149274804545598539851125960, 8.929290858322258575439899443543, 10.63146600654926412456720698967, 11.98463346042085617267739947882, 12.68075016279508411089530745401
